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Full text of "A Simple Derivation of the Equations of Einstein's Theory of Special Relativity"

A Simple Derivation of the Equations of Einstein's 
Theory of Special Relativity 

P.S.C. Bruskiewich 
Mathematical-Physics, University of British Columbia, Vancouver, BC 

The implications of the two simple postulates of Special Relativity proposed by Albert 
Einstein are presented in a succinct form in this paper. The simple algebraic derivation 
of the equations of Special Relativity was shared by the author to Dr. F. A. Kaempffer 
and Dr. George Volkoff during the course of a pleasant conversation in 1984. 



1.0 Einstein's two Simple Postulates of Special Relativity 

As proposed by Albert Einstein, there are two simple postulates in Special Relativity 

i) There is no preferred inertial frame, 

ii) The fasted a signal can travel in a frame of reference is the speed of light 

The implications of these two simple postulates is the Lorentz transformation which can 
be easily derived using simple algebra. 

2.0 The Lorentz Transformation 

Consider a transformation from a rest frame to a moving frame (the moving frame is 
denoted by prime), 



x = /(jc' + vf') 
x = y(x-vt) 



Evidentially if one frame is moving to the right with respect to the other, the other frame 
is moving to the left with respect to the first frame of reference. 
If a light beam is moving in both frames this 

x-ct 
x' = ct f 



From this we find 



ct = y (ct' + vf') =^> ct = y (c + v) t' 
ct' = y (ct - vt) ^> ct' = y (c - v) t 



If we multiply the two right hand equations together we find 



ct(ct') = y (c + v)t'y(c-v)t^>c 2 =^ 2 (c 2 -v 2 ) 



From whence we derive the Lorentz Transformation 



7 = 



1 



'1- 



The two equations then become 



V 



c 



x 



x 




(x' + vf) 



(x-vtj 



3.0 Einstein's Third Postulate of Special Relativity 

Notice how one manipulates the fundamental equations of motion in Special Relativity. 
This leads to a Third tacit or unwritten Postulate: 

iii) to go from one equation in one frame of reference to an equation in the other frame of 
reference, the prime variables become un-primed and vice versa and the velocity merely 
changes sign. 

The Third of Einstein's postulate is evidently a heuristic postulate. 

4.0 Einstein's Fourth Postulate of Special Relativity 

The functional form of the Lorentz transformation is the simplest function that provides 
for a transformation based on the dynamics of the problem. This leads to a fourth 
heuristic Postulate: 

iv) It is understandable that the Lorentz transformation involves both v and c because 
these are the only velocities observers in the two frames of reference can agree upon. 

5.0 Deriving the Time Transformation Equation 

The derivation of the time transformation involves two steps. 

The first step is to eliminate x in the transformation expression for x' , namely 

x = y(x- vt) => x = y(y(x + vt')- vt) 



The second step involves some basic algebra and simplification and solving for t, which 
yields 



t = r 



t + — X 



By Einstein's Third Postulate we find then that 



f 



t = r 



\ 



t + — X 
c J 



r 



ot =y 



\ 



v 

t y x 
V c J 



6.0 The Relativistic addition of velocities 



Using the definition of velocity 



Ax_ y(Ax' + vAt') 



At 



v 



\ 



/I At' + —r Ax' 



c 



J 



/At' 



(Ax' 



\ 



V 



At' 



+ v 



J 



yAt'\ 1 + 



v Ax 



'A 



c 2 At' 



J 



(u' + v) 
c 2 J 



By Einstein's Third Postulate we find then that 



(u' + v) , (u—v) 

U = -H ^r OU = 




1_ 7 M 



The value of the third and fourth of Einstein's heuristic postulates are self-evident. 
Presented in this fashion, the equations of Special Relativity can be understood by an 
individual with a good understanding of algebra. 

Acknowledgements: 

The author would like to posthumously thank Dr. F.A. Kaempffer and Dr. G.M. Volkoff 
for their discussion and encouragement in the writing of this short article. This paper was 
originally written in 1984 with their kind assistance. The manuscript remained lost in 
the author's papers for several decades, only recently rediscovered and published. 

© Patrick Bruskiewich 2103